Small Jump with Negation-UTM Trampoline

arXiv:1112.0987v17 [cs.CC] 11 Oct 2014 SMALL JUMP WITH NEGATION-UTM TRAMPOLINE KOJI KOBAYASHI 1. Introduction This paper divide some complexity class by using fixpoint and fixpointless area of Decidable Universal Turing Machine (UTM). Decidable Deterministic Turing Machine (DTM) have fixpointless combinator that add no extra resources (like Negation), but UTM makes some fixpoint in the combinator. This means that we can jump out of the fixpointless combinator system by making more complex problem from diagonalisation argument of UTM. As a concrete example, we proof L is not P . We can make Polynomial time UTM that emulate all Logarithm space DTM (LDTM). LDTM set close under Negation, therefore UTM does not close under LDTM set. (We can proof this theorem like halting problem and time/space hierarchy theorem, and also we can extend this proof to divide time/space limited DTM set.) In the same way, we proof P is not NP. These are new hierarchy that use UTM and Negation. 2. L is not P Definition 1. “DT M” is defined as Decidable Deterministic Turing Machine set. “LDT M” is defined as logarithmic space DT M. “pDT M” is defined as polynomial time DT M. “⃝DT M” is defined as DT M that some resource (time, space) limited. “UT M” is defined as Universal Turing Machine set. “UT M (C)” is defined as minimum UT M that can emulate all M ∈C. ⟨M⟩is defined as code number of a M ∈DT M that U ∈UT M emulate. That is, ∀w [U (⟨M⟩, w) = M (w)] and U (⟨M⟩) = M. “Negate (C)” is defined as minimum Negation system that include C. That is, ∀C [(C ⊂Negate (C)) ∧(∀c ∈Negate (C) [¬c ∈Negate (C)])]. Theorem 2. ∀r ∈⃝DT M (¬r ∈⃝DT M) Proof. It is trivial from DTM structure. If DTM is M = (Q, Σ, Γ, δ, q0, q1, q2) then this dual machine M = (Q, Σ, Γ, δ, q0, q2, q1) compute ¬M without extra resources. Therefore negation of ⃝DT M is also in ⃝DT M. □ Theorem 3. ∃U ∈UT M (LDT M)[U ∈pDT M] Proof. It is trivial because some U ′ ∈UT M can emulate all LDT M in polynomial time. Therefore, we can make U ∈pDT M by limiting at polynomial time (if U ′ compute over polynomial time, U reject these input). □ 1 SMALL JUMP WITH NEGATION-UTM TRAMPOLINE 2 Theorem 4. L ⊊P Proof. We can proof this theorem like halting problem and time/space hierarchy theorem. Because of ∀U ∈UT M (LDT M), M ∈LDT M [U (⟨M⟩) = M] 1 all M ∈LDT M have index ⟨M⟩. Therefore we can make H which is diagonal- ization of U. H (⟨M⟩) = U (⟨M⟩, ⟨M⟩) ⟨M0⟩ ⟨M1⟩ ⟨M2⟩ ⟨M3⟩ · · · M0 = { ⊤ ⊤ ⊥ ⊤ · · · M1 = { ⊥ ⊥ ⊤ ⊥ · · · M2 = { ⊥ ⊥ ⊥ ⊤ · · · M3 = { ⊤ ⊤ ⊥ ⊥ · · · ... ... ... ... ... H = { ⊤ ⊥ ⊥ ⊥ · · · H ∈pDT M because U ∈pDT M 2 and H input size is at least half of U. Mentioned above 2, ∀r ∈LDT M (¬r ∈LDT M) we can make G which is Negation of diagonalization. G (⟨M⟩) = ¬H (⟨M⟩) = ¬U (⟨M⟩, ⟨M⟩) ⟨M0⟩ ⟨M1⟩ ⟨M2⟩ ⟨M3⟩ · · · M0 = { ⊤ ⊤ ⊥ ⊤ · · · M1 = { ⊥ ⊥ ⊤ ⊥ · · · M2 = { ⊥ ⊥ ⊥ ⊤ · · · M3 = { ⊤ ⊤ ⊥ ⊥ · · · ... ... ... ... ... H = { ⊤ ⊥ ⊥ ⊥ · · · G = { ⊥ ⊤ ⊤ ⊤ · · · G /∈LDT M because ∀M ∈LDT M [G (⟨M⟩) ̸= M (⟨M⟩)]. On the other hand, G ∈pDT M because H ∈pDT M. Therefore, G ∈pDT M (G /∈LDT M) and L ⊊P. □ We can expand above result to general DTM. Theorem 5. ∀CC ⊂DT M [Negate (UT M (Negate (CC))) ⊈Negate (CC)] Proof. We omit the proof because this proof is same as previous. □ Corollary 6. Negate (UT M (⃝DT M)) ⊈⃝DT M 3. P is not NP Theorem 7. ∃U ∈UT M (pDT M)[U ∈pNT M] Proof. We can make some oracle TM which oracle emulate transition function. npp | np ∈NP, p ∈P p (⟨t⟩, w) = t (w) t ∈P: transition function ⟨t⟩:code number of transition function t w: t’s input (state and symbol) SMALL JUMP WITH NEGATION-UTM TRAMPOLINE 3 p (⟨t⟩, w) accept if and only if t (w) accept and output t (w). Oracle TM npp ∈pNT M and npp can emulate all pDT M. Therefore npp ∈ UT M (pDT M). □ Note 8. npp can change transition function more flexible than pDT M in less time. In fact, npp can increase transition function with logarithm time (by computing these transition functions in parallel). Therefore, npp have more chance to compute more complex problems. Theorem 9. P ⊊NP Proof. (Proof by contradiction.) Assume to the contrary that P = NP. P = NP means that P = NP = coNP = PH, therefore NP close under Negation. In the same way as mentioned above 4, we get P ⊊NP. This result contradicting assumption P = NP. Therefore P ⊊NP. □ Corollary 10. P ⊊coNP 4. Trampoline Hierarchy between Negation and UTM This result shows that we can jump over border of asymptotic analysis by using Negation (fixpointless combinator) and UTM (fixpoint creator). Therefore, combi- nation of UTM and Negation make new complexity class. That is, there are some Hierarchy of UTM and Negation. References [1] Michael Sipser, (translation) Kazuo OHTA, Keisuke TANAKA, Masayuki ABE, Hiroki UEDA, Atsushi FUJIOKA, Osamu WATANABE, “Introduction to the Theory of COMPUTATION Second Edition (Japanese version)”, 2008

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